Projective Representations

Quantum symmetries act on physical rays. Because a ray is unchanged by an overall phase, a symmetry group can be represented on Hilbert-space vectors only up to phases. Such a representation is called projective. It should be compared with an ordinary representation, where the group multiplication law is implemented exactly by linear operators.

For the pure-state space of rays itself, see Projective Hilbert Space.

Rays Allow Phase Freedom

If two normalized vectors differ by a global phase,

$$\lvert\psi'\rangle = e^{i\alpha}\lvert\psi\rangle,$$

they represent the same pure state. Therefore a symmetry operation only has to act consistently on rays, not on a specific phase choice for every vector.

This is the origin of projective representations in quantum mechanics.

Projective Composition Law

For a group $G$, an ordinary unitary representation satisfies

$$U(g)U(h)=U(gh).$$

A projective representation allows a phase:

$$U(g)U(h) = e^{i\alpha(g,h)}U(gh).$$

The phase factor does not change the physical ray, but it can have real structural consequences. Not every phase can be removed by redefining the phases of the individual $U(g)$.

Spin and Rotations

The most important example in elementary quantum mechanics is spin. Ordinary spatial rotations form $SO(3)$, but spin-$1/2$ states transform naturally under $SU(2)$, the double cover of $SO(3)$. The group-level comparison is developed in SU(2) versus SO(3).

For the distinction between a harmless global sign on one ray and a measurable relative sign in interference, see Spinors and 2π Rotations.

A spin-$1/2$ rotation is represented by

$$U(\hat{\mathbf n},\theta) = \exp\left( -\frac{i}{2}\theta\,\hat{\mathbf n}\cdot\boldsymbol\sigma \right).$$

A $2\pi$ rotation gives

$$U(2\pi)=-I.$$

This changes the vector but not the ray. A $4\pi$ rotation returns the vector itself.

Physical Meaning of the Phase

The statement “global phase is unobservable” is true for a single isolated state. It does not mean phases in symmetry composition are irrelevant. Projective phases determine which Hilbert-space representations are possible and how states transform under combined operations.

They are especially important in:

• spinor representations of rotations,
• magnetic translations,
• Galilean boosts and mass as a central charge,
• anyon statistics and topological phases in more advanced settings.

Rephasing and Convention Dependence

Changing representatives by phases,

$$U(g)\longmapsto e^{i\beta(g)}U(g),$$

changes the function $\alpha(g,h)$ but not the physical projective action on rays. The invariant question is whether all such phases can be removed globally. If not, the projective representation carries nontrivial information.

Boundary of This Page

This page gives the quantum-mechanical motivation and the spin bridge. A full mathematical classification of projective representations belongs in a group-theory or mathematical physics treatment. Later spin pages use this result rather than reproving it.

Common Mistakes

• Thinking projective means “approximate” or “not really a representation.”
• Forgetting that the physical state is a ray.
• Treating the spinor sign change under $2\pi$ rotation as a contradiction.
• Assuming all phase factors can always be removed by convention.
• Confusing the group acting on physical space, $SO(3)$, with its double cover, $SU(2)$.

Cross-Links

• Rays and Global Phase
• Projective Hilbert Space
• Quantum Symmetries
• Unitary Symmetries
• Symmetry Groups and Representations
• Galilean Boosts
• Representations
• Unitary Representations
• Notation and Conventions
• SO(3)
• SU(2)
• SU(2) versus SO(3)
• SO(3) and SU(2) Preview
• Spinors and 2π Rotations
• From Spin to Relativistic Representations
• From SU(2) Spinors to Lorentz Spinors
• From Projective Representations to Anomalies Preview
• Pauli Matrices

References

• V. Bargmann, “On unitary ray representations of continuous groups,” Annals of Mathematics 59, 1–46, 1954.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Springer, 2015.

Exercises

1. Show that $U(2\pi)=-I$ for a spin-$1/2$ rotation about the $z$ axis.

Solution

For spin-$1/2$,

$$U(\hat z,\theta) = \exp\left(-\frac{i}{2}\theta\sigma_z\right).$$

For $\theta=2\pi$,

$$U(\hat z,2\pi)=e^{-i\pi\sigma_z}.$$

Since $\sigma_z$ has eigenvalues $\pm1$, both exponential eigenvalues are $-1$. Thus $U(\hat z,2\pi)=-I$.

2. Why does a projective phase in $U(g)U(h)=e^{i\alpha(g,h)}U(gh)$ not change transition probabilities?

Solution

The phase multiplies the transformed vector by an overall unit complex number. Transition probabilities use absolute squares of inner products, so the phase cancels with its complex conjugate.